Fredholm index of Toeplitz pairs with $H^{\mathrm{\infty }}$ symbols

In the present paper, we characterize the Fredholmness of Toeplitz pairs on Hardy space over the bidisk with the bounded holomorphic symbols, and hence we obtain the index formula for such Toeplitz pairs. The key to obtain the Fredholmness of such Toeplitz pairs is the $L^p$ solution of Corona Problem over ${\mathbb{D}}^2$.